Question

The function f(x) (2x + 1)(x - 2)² has a turning point at x = 2, which of the following statements are true? Select one: O a. f(x) has a maximum at the turning point. O b. f(x) has no tangent at the turning point. O c. f(x) has a point of inflection at the turning point. O d. f(x) is undefined at the turning point. O e. f(x) has a minimum at the turning point.

Answer 1

Therefore, the statement "f(x) has a minimum at the turning point" is true.

Explanation:

The correct statement is:

O e. f(x) has a minimum at the turning point.

To determine the nature of the turning point at x = 2, we can analyze the behavior of the function f(x) = (2x + 1)(x - 2)² in the vicinity of x = 2.

When a quadratic factor (x - 2)² is multiplied by a linear factor (2x + 1), the turning point occurs at the value of x that makes the linear factor equal to zero. In this case, when 2x + 1 = 0, we find x = -1/2. This is the x-coordinate of the turning point.

Now, we need to determine whether the turning point is a minimum or maximum. To do this, we can examine the behavior of the quadratic factor (x - 2)².

Since (x - 2)² is squared, it is always non-negative or zero. When x = 2, the quadratic factor is equal to zero, indicating that the turning point is located at the minimum of the function. Therefore, the statement "f(x) has a minimum at the turning point" is true.